I'm wondering what is the right way to perform function composition on those two functions:
$$f\left(x\right) = \left\{ \begin{array}{lr} 1/x & : x \ne 0\\ 0 & : x = 0 \end{array} \right.\\$$
$$g\left(t\right)=1/t$$
I want to create $f∘g$ and see what is the end result of the composition. I know that it should be easy but I'm a little bit confused determining the domain and range of the new function.
Help would be appreciated!
Thank you.
You want to calculate $f(g(t)) = f(1/t)$. We have $$ f(1/t) = \cases{\frac1{1/t} = t& if $1/t \neq 0$\\0 & if $1/t = 0$} $$ which is to say $f(g(t)) = t$. Note that since $g$ is undefined for $t = 0$, so is $f \circ g$. The function $f$ is defined for any real number, so there is no specific value of $t$ that makes $g(t)$ inadmissable as argument to $f$. The domain is therefore $\Bbb R \setminus \{0\}$